For any subset \(\theta \subset \mathbb {R}^c\) , assume that \({S}(\theta )\)  represents the space of Borel probability measures on \(\theta \) . The Wasserstein space
of order \(k \in [1, \infty )\)  of probability measures on \(\theta \)  is defined as follows: \({S}_k(\theta )=\lbrace  \mathcal {F} \in {S}(\theta ): \int ||x||^k d \mathcal {F}(x) < \infty \rbrace \) , where, \(||.||\)  is the Euclidean distance in \(\mathbb {R}^c\) . Let \({\Pi } (\mathcal {P}, \mathcal {Q})\)  denote the set of all probability measures on
\(\theta \times \theta \)  which have marginals \(\mathcal {P}\)  and \(\mathcal {Q}\) ; then the \(k\) -th Wasserstein distance between \(\mathcal {P}\)  and \(\mathcal {Q}\)  in \({S}_k(\theta )\) , is defined as follows [1]}:
\(W_k(\mathcal {P}, \mathcal {Q})=\bigg (\inf _{\pi \in {\Pi (\mathcal {P}, \mathcal {Q})}} \int _ {\theta ^2} ||x-x^{\prime }||^k d\pi (x,x^{\prime }) \bigg )^{{1}/{k}},\) 
